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12
Preliminaries
10101Basicfactsonidealsandfilters
LetRbeanonemptyset.AnidealIonasetRisacollectionsuchthat
(1)∅∈I;
(2)ifA,B∈IthenA∪B∈I;
(3)ifA,B⊂R,A∈IandB⊂AthenB∈I.
Inthewholepapertherewillbeonlyconsideredproperidealsi.e.,R̸∈I.
Moreover,thenotionA∈I+meansA̸∈Ii.e.,I+=P(R)\I.
AfilterFonasetRisacollectionsuchthat
(1)R∈F;
(2)ifA,B∈FthenA∩B∈F;
(3)ifA,B⊂R,A∈FandA⊂BthenB∈F.
IfIisanidealthen
F={R\A:A∈I}
isafilter.IfFisafilterthen
I={R\A:A∈F}
isanideal.InthissituationFandIarecalleddual.
Letλ⩾ωbeacardinal.Then
[λ]<λ={A⊂λ:|A|<λ}
iscalledtheFréchetidealonλ.
LetAobeanonemptysubsetofR.Thefilter
F={A⊂R:A⊃Ao}
iscalledaprincipalfilter.Thedualnotionisaprincipalideal.
AnidealIiscalledcountablycomplete(oraσ-ideal)ifU
n∈ωAn∈I
wheneverAn∈Iforanyn∈ω.
